Scattering and spectral theory for manifolds

摘要：dms -0504729首席研究员：Harold donnelli首席研究员计划研究三个独立的主题：i)共振计数函数的下界，ii)量子唯一遍历性，iii)双曲空间理想边界附近特征函数的行为。奇维欧几里得空间共振计数函数的已知上界与下界存在较大的差异。据推测，存在一个与已知上界相当的一般下界。建议构造反例来证伪该假设。量子唯一遍历性涉及遍历测地线流流形上特征函数的集中。在早期的工作中，构造了违反本征函数包的量子唯一遍历性的流形。现在提出了构造单个特征函数的量子唯一遍历性的反例。所提出的例子将是具有旋转对称性的流形的扰动，在例外情况下，KAM理论不能提供摄动系统的不变性。双曲空间的拉普拉斯算子具有绝对连续的谱。如果改变紧子集上的度规，就可能存在特征值位于本质谱底部以下的特征函数。我们的目标是理解在无穷远处理想边界附近这些特征函数的节点集。共振是与原子光谱有关的物理现象的数学模型。一般来说，如果不受扰动，状态不会永远存在，共振对应于振荡的衰变状态。数学家试图通过与经典力学的类比来理解量子力学。如果经典运动是混沌的，人们期望量子粒子的概率分布是分散的。这种模式的例外，粒子集中的地方，是特别有趣的。量子粒子的节点集是相关波情感的平稳集。它的分布和形状是最重要的。

AbstractAward: DMS-0504729Principal Investigator: Harold DonnellyThe principal investigator plans to study three separate topics:i) Lower bounds on the counting function for resonances, ii)Quantum unique ergodicity, iii) Behavior of eigenfunctions nearthe ideal boundary of hyperbolic space.There is a largediscrepancy between the known upper and lower bounds for theresonance counting function of Euclidean space with odddimension.It has been conjectured that there is a general lowerbound which is comparable to the known upper bound.The proposalis to construct counterexamples which falsify theconjecture.Quantum unique ergodicity concerns concentration ofeigenfunctions on manifolds with ergodic geodesic flow.In earlierwork,manifolds were constructed where quantum unique ergodicityis violated for packets of eigenfunctions.It is now proposed toconstruct counterexamples to quantum unique ergodicity forindividual eigenfunctions.The proposed examples would beperturbations of manifolds with rotational symmetry,in theexceptional cases where KAM theory fails to provide invarianttori for the perturbed system.The Laplacian of hyperbolic spacehas absolutely continuous spectrum.If one alters the metric on acompact subset there may exist eigenfunctions whose eigenvalueslie below the bottom of the essential spectrum.The goal is tounderstand the nodal set of these eigenfunctions near the idealboundary at infinity.Resonances are mathematical models for physical phenomena relatedto atomic spectra.In general,states do not exist forever ifunperturbed and resonances correspond to decaying states whichoscillate.Mathematicians seek to understand quantum mechanics byanalogy with classical mechanics.If the classical motion ischaotic,one expects the probability distribution of the quantumparticle to be dispersed.Exceptions to this pattern,where theparticle concentrates,are of particular interest.The nodal set ofa quantum particle is the stationary set for the associated wavemotion.Its distribution and shape are of basic importance.